Copyright © 2011 Pearson, Inc.

4.3Trigonometry

Extended: The Circular

Functions

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What you’ll learn about

Trigonometric Functions of Any Angle Trigonometric Functions of Real Numbers Periodic Functions The 16-point unit circle

… and whyExtending trigonometric functions beyond triangle ratios opens up a new world of applications.

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Initial Side, Terminal Side

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Positive Angle, Negative Angle

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Coterminal Angles

Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles.

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Example Finding Coterminal Angles

Find a positive angle and a negative angle that are

coterminal with 45º .

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Example Finding Coterminal Angles

Add 360º: 45º+360º=405º

Subtract 360º: 45º−360º=−315º

Find a positive angle and a negative angle that are

coterminal with 45º .

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Example Finding Coterminal Angles

Find a positive angle and a negative angle that are

coterminal with π6.

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Example Finding Coterminal Angles

Find a positive angle and a negative angle that are

coterminal with π6.

Add 2π :π6+ 2π =

13π6

Subtract 2π :π6−2π =−

11π6

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Example Evaluating Trig Functions Determined by a Point in QI

Let θ be the acute angle in standard position whose

terminal side contains the point (3,5).Find the six trigonometric functions of θ.

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Example Evaluating Trig Functions Determined by a Point in QI

Let θ be the acute angle in standard position whose

terminal side contains the point (3,5).Find the six trigonometric functions of θ.

The distance from (3,5) to the origin is 34.

sinθ =534

≈0.857 cscθ =345

≈1.166

cosθ =334

≈0.514 secθ =343

≈1.944

tanθ =53 cotθ =

35

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Trigonometric Functions of any Angle Let θ be any angle in standard position and let P(x,y)

be any point on the terminal side of the angle (except

the origin). Let r denote the distance from P(x,y)

to the origin, i.e., let r = x2 + y2 . Then

sinθ =yr cscθ =

ry (y≠0)

cosθ =xr secθ =

rx (x≠0)

tanθ =yx (x≠0) cotθ =

xy (y≠0)

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Evaluating Trig Functions of a Nonquadrantal Angle θ

1. Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant.

2. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ.

3. Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator.

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Evaluating Trig Functions of a Nonquadrantal Angle θ

4. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant.

5. Use the coordinates of point P and the definitions to determine the six trig functions.

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Example Evaluating More Trig Functions

Find sin 210º without a calculator.

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Example Evaluating More Trig Functions

Find sin 210º without a calculator.

An angle of 210º in standard position determines

a 30º–60º–90º reference triangle in the third quadrant.

The lengths of the sides stermines the point P − 3,−1( ).

The hypotenuse is r =2.

sin210º=yr=−

12

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Example Using one Trig Ration to Find the Others

Find sinθ and cotθ by using the given information toconstruct a reference triangle.

a. cosθ =−817

and cscθ < 0

b. tanθ =−12 and cosθ > 0

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Example Using one Trig Ration to Find the Others

Since cosθ < 0 and cscθ =1

sinθ< 0

the terminal side is in QIII.Draw a reference triangle with

r =17, x=−8.

and y= 172 −82 =−15

sinθ =−1517

and cotθ =815

a. cosθ =−817

and cscθ < 0

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Example Using one Trig Ration to Find the Others

Since tanθ < 0 and cosθ > 0, the terminal side is in QIV.Draw a reference triangle with

x=2, y=−1.

and r = 22 +12 = 5

sinθ =−15≈-0.447 and

cotθ =−21=−2

b. tanθ =−12 and cosθ > 0

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Unit Circle

The unit circle is a circle of radius 1 centered at the origin.

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Trigonometric Functions of Real Numbers

Let t be any real number, and let P(x, y) be the point

corresponding to t when the number line is wrapped

onto the unit circle as described above. Then

sin t =y csct=1y (y≠0)

cost=x sect=1x (x≠0)

tant=yx (x≠0) cott=

xy (y≠0)

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Periodic Function

A function y = f (t) is periodic if there is a positive

number c such that f (t+ c) = f (t) for all values of

t in the domain of f . The smallest such number cis called the period of the function.

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The 16-Point Unit Circle

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Quick Review

Give the value of the angle θ in degrees.

1. θ =2π3

2. θ =−π4

Use special triangles to evaluate.

3. cot −π4

⎛⎝⎜

⎞⎠⎟

4. cos −7π6

⎛⎝⎜

⎞⎠⎟

5. Use a right triangle to find the other five trigonometric

functions of the acute angle θ given cosθ =45

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Quick Review Solutions

Give the value of the angle θ in degrees.

1. θ =2π3 120o 2. θ =−

π4 −45o

Use special triangles to evaluate.

3. cot −π4

⎛⎝⎜

⎞⎠⎟ −1 4. cos −

7π6

⎛⎝⎜

⎞⎠⎟ − 3 / 2

5. Use a right triangle to find the other five trigonometric

functions of the acute angle θ given cosθ =45

secθ =5 / 4, sinθ =3 / 5, cscθ =5 / 3,tanθ =3 / 4, cotθ =4 / 3